Properties

Label 100.9.d.b
Level $100$
Weight $9$
Character orbit 100.d
Analytic conductor $40.738$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [100,9,Mod(99,100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.99");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 100.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(40.7378610061\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{39})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 19x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 4)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} - 8 \beta_1 q^{3} + ( - 2 \beta_{3} + 56) q^{4} + (8 \beta_{3} - 1248) q^{6} + 112 \beta_1 q^{7} + ( - 368 \beta_{2} - 144 \beta_1) q^{8} + 3423 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} - 8 \beta_1 q^{3} + ( - 2 \beta_{3} + 56) q^{4} + (8 \beta_{3} - 1248) q^{6} + 112 \beta_1 q^{7} + ( - 368 \beta_{2} - 144 \beta_1) q^{8} + 3423 q^{9} + 148 \beta_{3} q^{11} + (2496 \beta_{2} - 448 \beta_1) q^{12} + 547 \beta_{2} q^{13} + ( - 112 \beta_{3} + 17472) q^{14} + ( - 224 \beta_{3} - 59264) q^{16} + 7309 \beta_{2} q^{17} + ( - 3423 \beta_{2} + 3423 \beta_1) q^{18} - 156 \beta_{3} q^{19} - 139776 q^{21} + (23088 \beta_{2} + 14800 \beta_1) q^{22} + 18992 \beta_1 q^{23} + (2944 \beta_{3} + 179712) q^{24} + (547 \beta_{3} + 54700) q^{26} + 25104 \beta_1 q^{27} + ( - 34944 \beta_{2} + 6272 \beta_1) q^{28} + 128222 q^{29} + 544 \beta_{3} q^{31} + (24320 \beta_{2} - 81664 \beta_1) q^{32} - 184704 \beta_{2} q^{33} + (7309 \beta_{3} + 730900) q^{34} + ( - 6846 \beta_{3} + 191688) q^{36} - 347203 \beta_{2} q^{37} + ( - 24336 \beta_{2} - 15600 \beta_1) q^{38} - 4376 \beta_{3} q^{39} + 2146882 q^{41} + (139776 \beta_{2} - 139776 \beta_1) q^{42} + 474632 \beta_1 q^{43} + (8288 \beta_{3} + 4617600) q^{44} + ( - 18992 \beta_{3} + 2962752) q^{46} + 610592 \beta_1 q^{47} + (279552 \beta_{2} + 474112 \beta_1) q^{48} - 3807937 q^{49} - 58472 \beta_{3} q^{51} + (30632 \beta_{2} + 109400 \beta_1) q^{52} - 82429 \beta_{2} q^{53} + ( - 25104 \beta_{3} + 3916224) q^{54} + ( - 41216 \beta_{3} - 2515968) q^{56} + 194688 \beta_{2} q^{57} + ( - 128222 \beta_{2} + 128222 \beta_1) q^{58} - 29828 \beta_{3} q^{59} - 14746078 q^{61} + (84864 \beta_{2} + 54400 \beta_1) q^{62} + 383376 \beta_1 q^{63} + (105984 \beta_{3} - 10307584) q^{64} + ( - 184704 \beta_{3} - 18470400) q^{66} + 1221512 \beta_1 q^{67} + (409304 \beta_{2} + 1461800 \beta_1) q^{68} - 23702016 q^{69} + 9576 \beta_{3} q^{71} + ( - 1259664 \beta_{2} - 492912 \beta_1) q^{72} + 572563 \beta_{2} q^{73} + ( - 347203 \beta_{3} - 34720300) q^{74} + ( - 8736 \beta_{3} - 4867200) q^{76} + 2585856 \beta_{2} q^{77} + ( - 682656 \beta_{2} - 437600 \beta_1) q^{78} + 287536 \beta_{3} q^{79} - 53788095 q^{81} + ( - 2146882 \beta_{2} + 2146882 \beta_1) q^{82} + 4160152 \beta_1 q^{83} + (279552 \beta_{3} - 7827456) q^{84} + ( - 474632 \beta_{3} + 74042592) q^{86} - 1025776 \beta_1 q^{87} + ( - 3324672 \beta_{2} + 5446400 \beta_1) q^{88} + 83324222 q^{89} + 61264 \beta_{3} q^{91} + ( - 5925504 \beta_{2} + 1063552 \beta_1) q^{92} - 678912 \beta_{2} q^{93} + ( - 610592 \beta_{3} + 95252352) q^{94} + ( - 194560 \beta_{3} + 101916672) q^{96} + 12061901 \beta_{2} q^{97} + (3807937 \beta_{2} - 3807937 \beta_1) q^{98} + 506604 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 224 q^{4} - 4992 q^{6} + 13692 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 224 q^{4} - 4992 q^{6} + 13692 q^{9} + 69888 q^{14} - 237056 q^{16} - 559104 q^{21} + 718848 q^{24} + 218800 q^{26} + 512888 q^{29} + 2923600 q^{34} + 766752 q^{36} + 8587528 q^{41} + 18470400 q^{44} + 11851008 q^{46} - 15231748 q^{49} + 15664896 q^{54} - 10063872 q^{56} - 58984312 q^{61} - 41230336 q^{64} - 73881600 q^{66} - 94808064 q^{69} - 138881200 q^{74} - 19468800 q^{76} - 215152380 q^{81} - 31309824 q^{84} + 296170368 q^{86} + 333296888 q^{89} + 381009408 q^{94} + 407666688 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 19x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 29\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 9\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 40\nu^{2} - 380 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 5\beta_1 ) / 20 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 380 ) / 40 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 29\beta_{2} + 45\beta_1 ) / 20 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(77\)
\(\chi(n)\) \(-1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
−3.12250 + 0.500000i
−3.12250 0.500000i
3.12250 + 0.500000i
3.12250 0.500000i
−12.4900 10.0000i 99.9200 56.0000 + 249.800i 0 −1248.00 999.200i −1398.88 1798.56 3680.00i 3423.00 0
99.2 −12.4900 + 10.0000i 99.9200 56.0000 249.800i 0 −1248.00 + 999.200i −1398.88 1798.56 + 3680.00i 3423.00 0
99.3 12.4900 10.0000i −99.9200 56.0000 249.800i 0 −1248.00 + 999.200i 1398.88 −1798.56 3680.00i 3423.00 0
99.4 12.4900 + 10.0000i −99.9200 56.0000 + 249.800i 0 −1248.00 999.200i 1398.88 −1798.56 + 3680.00i 3423.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.9.d.b 4
4.b odd 2 1 inner 100.9.d.b 4
5.b even 2 1 inner 100.9.d.b 4
5.c odd 4 1 4.9.b.b 2
5.c odd 4 1 100.9.b.c 2
15.e even 4 1 36.9.d.b 2
20.d odd 2 1 inner 100.9.d.b 4
20.e even 4 1 4.9.b.b 2
20.e even 4 1 100.9.b.c 2
40.i odd 4 1 64.9.c.b 2
40.k even 4 1 64.9.c.b 2
60.l odd 4 1 36.9.d.b 2
80.i odd 4 1 256.9.d.e 4
80.j even 4 1 256.9.d.e 4
80.s even 4 1 256.9.d.e 4
80.t odd 4 1 256.9.d.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.9.b.b 2 5.c odd 4 1
4.9.b.b 2 20.e even 4 1
36.9.d.b 2 15.e even 4 1
36.9.d.b 2 60.l odd 4 1
64.9.c.b 2 40.i odd 4 1
64.9.c.b 2 40.k even 4 1
100.9.b.c 2 5.c odd 4 1
100.9.b.c 2 20.e even 4 1
100.9.d.b 4 1.a even 1 1 trivial
100.9.d.b 4 4.b odd 2 1 inner
100.9.d.b 4 5.b even 2 1 inner
100.9.d.b 4 20.d odd 2 1 inner
256.9.d.e 4 80.i odd 4 1
256.9.d.e 4 80.j even 4 1
256.9.d.e 4 80.s even 4 1
256.9.d.e 4 80.t odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 9984 \) acting on \(S_{9}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 112 T^{2} + 65536 \) Copy content Toggle raw display
$3$ \( (T^{2} - 9984)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 1956864)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 341702400)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 29920900)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 5342148100)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 379641600)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 56268585984)^{2} \) Copy content Toggle raw display
$29$ \( (T - 128222)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4616601600)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 12054992320900)^{2} \) Copy content Toggle raw display
$41$ \( (T - 2146882)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 35142983526144)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 58160324112384)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 679454004100)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 13879469510400)^{2} \) Copy content Toggle raw display
$61$ \( (T + 14746078)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 232766284318464)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 1430516505600)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 32782838896900)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 26\!\cdots\!24)^{2} \) Copy content Toggle raw display
$89$ \( (T - 83324222)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
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